Here we calculate the theoretical GRB neutrino flux according to two models, the proton escape model [1]} and the neutron escape model [2]}. We assume that there are 667 GRBs all over the full sky per year. The baryonic loading, which means the ratio of fireball energy in protons to electrons, is set as \(f_p=10\) . The bulk Lorentz factor \(\Gamma \)  of the fireball is set as a benchmark value \(\Gamma =300\) , which leads to a double broken power laws of neutrino spectra peaking around 100 TeV [1]}. To compare the theoretical results with the measurements of the ultra-high energy neutrino flux, we introduce a generic double broken power-law neutrino flux [4]}:
\(\Phi _\nu (E_\nu )=\Phi _0 \times \left\lbrace \begin{array}{ll}\epsilon _b^{-1}E_\nu ^{-1}, & E_\nu \le \epsilon _b;\\E_\nu ^{-2}, & \epsilon _b< E_\nu \le 10\epsilon _b;\\E_\nu ^{-4}(10\epsilon _b)^2, & 10\epsilon _b< E_\nu ,\\\end{array}\right.\) 
